A force of \(20\) N acts on a particle along a direction, making an angle of \(60^\circ\) with the vertical. The component of the force along the vertical direction will be:

If $\overrightarrow{\mathrm{A}}=2\hat{\mathrm{i}}+4\hat{\mathrm{j}}-5\hat{\mathrm{k}}$, then the direction cosines of the vector are:

(direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three +ve coordinate axes.)

1. $\frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}} \mathrm{and} \frac{-5}{\sqrt{45}}$

2. $\frac{1}{\sqrt{45}}, \frac{2}{\sqrt{45}} \mathrm{and} \frac{3}{\sqrt{45}}$

3. $\frac{4}{\sqrt{45}}, 0 \mathrm{and} \frac{4}{\sqrt{45}}$

4. $\frac{3}{\sqrt{45}}, \frac{2}{\sqrt{45}} \mathrm{and} \frac{5}{\sqrt{45}}$

$\overrightarrow{\mathrm{A}}$ is a vector with magnitude A, then the unit vector $\hat{\mathrm{A}}$ in the direction of $\overrightarrow{\mathrm{A}}$ is

1. $\mathrm{A}\overrightarrow{\mathrm{A}}$

2. $\overrightarrow{\mathrm{A}}.\overrightarrow{\mathrm{A}}$

3. $\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{A}}$

4. $\overrightarrow{\mathrm{A}}/\mathrm{A}$

A force \(\mathrm{F}\) applied at a \(30^\circ\) angle to the \(\mathrm{x }\)-axis has the following X and Y components:

1. $\frac{\mathrm{F}}{\sqrt{2}}, \mathrm{F}$

2. $\frac{\mathrm{F}}{2}, \frac{\sqrt{3}}{2}\mathrm{F}$

3. $\frac{\sqrt{3}}{2}\mathrm{F}, \frac{1}{2}\mathrm{F}$

4. $\mathrm{F}, \frac{\mathrm{F}}{\sqrt{2}}$

A child pulls a box with a force of $200 \mathrm{N}$ at an angle of $60°$above the horizontal. Then the horizontal and vertical components of the force will be:
              

1. 100 N, 175 N

2.  86.6 N, 100 N  

3.  100 N, 86.6 N

4. 100 N, 0 N

If  $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=7\hat{i}+24\hat{k}$, then the vector having the same magnitude as that of $\overrightarrow{B}$ and parallel to $\overrightarrow{A}$ is:

1. $15\hat{i} + 20\hat{j}$

2. $\frac{7}{5}\hat{i}+\frac{24}{5}\hat{j}$

3. $20\hat{i}+15\hat{j}$

4. $15\hat{i} + 20\hat{k}$ 

A force is 60 ° inclined to the horizontal. If its rectangular component in the horizontal direction is 50 N, then the magnitude of the force in the vertical direction is

1. 25 N

2. 75 N

3. 87 N

4. 100 N

Figure shows the orientation of two vectors u and v in the XY plane.

$\text{ If }\mathrm{u}=\mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}\text{ and }\mathrm{v}=\mathrm{p}\hat{\mathrm{i}}+\mathrm{q}\hat{\mathrm{j}}$

Which of the following is correct?

1. a and p are positive while b and q are negative

2. a, p, and b are positive while q is negative

3. a, q, and b are positive while p is negative

4. a, b, p, and q are all positive

If a unit vector $\hat{j}$ is rotated through an angle of 45° anticlockwise, then the new vector will be:

1.  $\sqrt{2}\hat{\mathrm{i}} + \sqrt{2}\hat{\mathrm{j}}$

2.  $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

3.  $\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{\sqrt{2}}\hat{\mathrm{j}}$

4.  $-\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{\sqrt{2}}\hat{\mathrm{j}}$